Since we only extend \(\mathfrak {V}^{rb}\)  with approximations of the true solution snapshots of the full-order model,
note that Algorithm REF  is no longer a weak greedy algorithm in the sense of [1]}.
In particular note that the model reduction error is generally non-zero even for parameters \(\mu ^*\)  for which the
corresponding Stage 1 snapshot has been added to \(\mathfrak {V}^{rb}\) .
Thus, when \(\varepsilon _1\)  is chosen too large in comparison to \(\varepsilon _2\) , a single \(\mu ^*\)  might be selected
twice, causing Algorithm REF  to fail.
In such a case, the individual Stage 1 errors for each \(T \in \mathcal {T}_H\)  can be estimated to further enrich the Stage 1
spaces for which the error is too large.
We will not discuss such an approach in more detail here and instead note that in practice it is feasible to choose
\(\varepsilon _1\)  small enough to avoid such issues (cf. sec:experiments).
In particular, for sufficiently small \(\varepsilon _1\)  we can expect the convergence rates of a weak greedy algorithm
with exact solution snapshots to be preserved by Algorithm REF  up to the given target tolerance
\(\varepsilon _2\) .
